The principle of no equivalence: an agent-based model

2.1 Semantic differences

Two constructions may differ in their semantic value. The term ‘semantics’ is used in its most traditional sense as referring to propositional/truth-conditional meaning (though see Leclercq 2020 for a discussion on the use of semantics/pragmatics in Construction Grammar). There is little controversy about this claim. A large strand of research in lexical semantics has focused on finding out, among other factors, the semantic differences between pairs of near-synonyms (cf. Cruse 1986: 265–294). In Construction Grammar, focus on semantic differences was foundational to establishing the separate status of similar argument structure constructions. Examples (1) and (2) are cases in point.

The sentences in (1) and (2) respectively illustrate uses of the with-APPLICATIVE construction and the LOCATIVE CAUSED-MOTION construction, which constitute the locative alternation (Anderson 1971; Perek 2012). The meaning of these constructions is similar, but there is a key semantic difference: the with-APPLICATIVE construction focuses on the result state and entails a holistic reading (whereby the bread/face is fully covered with butter/shaving cream) while the LOCATIVE CAUSED-MOTION construction focuses on the action and makes a partial reading available (with only parts of the bread/face being covered with butter/shaving cream).

In Construction Grammar, alternating argument structure constructions are textbook examples given in support of the principle of no equivalence, especially with regard to semantic differences. These differences can be observed at every level of the constructicon however. For instance, semantic distinctions can be observed at the morphological level. Consider what is known as the spilled-spilt alternation (Faulkner 2023; Levin 2009; Quirk 1970; Rohdenburg 2003), which concerns a variety of verbs including burn, dream, dwell, kneel, lean, leap, learn, smell, spell, spill and spoil.

The -ed/-t alternation in the preterite (3) and participial (4) forms is said to reflect an aspectual distinction whereby the -ed variant is used with situations that are typically more durative (imperfective) while the -t variant is used with shorter (perfective) events (Levin 2009; Quirk 1970, also see De Smet and Van de Velde 2020; De Smet and Rosseel 2021 for Dutch). According to Bolinger (1968: 110; cited in Rohdenburg 2003), this difference is iconically motivated by the respective orthographic and phonetic length of the -ed and -t inflectional markers. Faulkner (2023) argues this difference carries over to the adjectival uses as well (5), whereby the -ed variant is perceived as adjustable (i.e., ‘not fully rotten’ in (5a)), while in comparison the -t variant refers to inalterable events (i.e., ‘toxic’ in (5b)). These examples show that the spilled-spilt alternation is not a case of free variation but that it is characterised by a semantic distinction.^[2] Similar observations can be made with more idiomatic (i.e., lexically fixed) constructions. For instance, Leclercq (2022: 241) discusses the semantic difference between the idioms ‘I can’t help V-ing’ (6) and ‘I can’t help but V’ (7).

These two constructions serve a very similar function since they are both “used by the speaker to express her inability not to perform the action denoted by the verbal complement” (Leclercq 2022: 242). The key difference though is that greater imposition is construed when using ‘I can’t help but V’ (largely due to the use of the negative polarity adverb but) which is not expressed with the gerund alternative ‘I can’t help V-ing’. So choosing one or the other construction largely depends on the speaker’s semantic intentions.

2.2 Pragmatic differences

The principle of no equivalence states that functional differences between two constructions may be found at the pragmatic level. As opposed to the notion of semantics, the label ‘pragmatics’ leaves more room for interpretation though, and a clear definition is needed. Goldberg (1995: 67) vaguely points out that pragmatic features include “particulars of information structure, including topic and focus, and additionally stylistic aspects of the construction such as register.” This illustrates what might be called a ‘broad’ view of pragmatics (Culpeper 2021), which includes any aspect of meaning that is not properly semantic (i.e., propositional): what Morin (2023) calls ‘circumstantial’ meaning. However, for the sake of clarity and precision in the study of synonymy avoidance, and to avoid any confusion entailed by a coarse-grained delineation of pragmatic meaning in no equivalence, we use the term ‘pragmatic’ in its ‘narrow’ acceptation, i.e., one that refers solely to utterance-focused features of meaning such as presuppositions, implicatures, illocutionary acts, speaker attitudes and information structure. Therefore, in our definition, features such as register fall outside the pragmatic domain into what we call the ‘social meaning’ of a construction (see Section 2.3 for details). Distinguishing between pragmatic and social meaning is as necessary as was reframing the principle in terms of no equivalence. This is because in both cases Goldberg (1995: 67) uses terminologically loaded terms – synonymy and pragmatics – in a much broader sense than is otherwise common practice in the literature. The shift from synonymy to equivalence enabled us to move away from the undesirable conclusion that the principle only pertains to semantics. It is similar concerns of terminological clarity and consistency that lead us to separate pragmatic and social meanings. Indeed, the latter type is not included in all definitions of pragmatics (cf. Ariel 2010; Horn and Ward 2004; Huang 2014, 2017; Levinson 1983), and we want to ensure that it be explicitly identified. Before presenting the exact scope of social meaning, this section provides a few examples of pragmatic differences.

In Construction Grammar, as exemplified in Goldberg’s quote above, information structure is treated as a prime example of pragmatic information (see Hilpert 2019; Leino 2013). This is what we first turn our attention to with examples of the COMPARATIVE CORRELATIVE (CC) construction (cf. Hoffmann 2019 and references cited therein).

This biclausal construction involves a covariational comparison between an independent variable (C1, e.g., the more connected we are) and a dependent variable (C2, e.g., the more time we need to ourselves). The examples in (8) illustrate the most canonical form of this construction (C1C2). But the order of the two clauses may also be reversed (C2C1), such as in (9). This difference involves other formal changes. For instance, while the C2 comparative phrase is clause-initial in C1C2 (e.g., the more successful you’re likely to be), it is post-verbal in C2C1 (e.g., the app becomes smarter). The differences in form between these two CC constructions reflects a functional difference: “while they are semantically synonymous, C1C2 and C2C1 constructions differ with respect to their information structure properties” (Hoffmann 2019: 157). Specifically, the C2C1 construction enables speakers to put focus on C2. This is why it is possible to use focus particles such as even in the C2C1 constructions, e.g., (9c), which is not felicitous in the C1C2 construction (e.g., ?The more connected we are, the more we even need time to ourselves). This difference in information structure is a typical pragmatic difference that can be found between (similar) constructions.

Pragmatic differences as specified by no equivalence can also materialise in other ways. A textbook example of pragmatic meaning are implicatures (Grice 1975; Zufferey et al. 2019). Leclercq and Depraetere (2022) argue that implicated meaning constitutes a key factor for the choice of be able to over can/could.

It is sometimes assumed that be able to is semantically synonymous with can and could and only serves as a morphosyntactic suppletive form, e.g., for future marking (Westney 1995: 209), see examples in (10). However, be able to is very often used in contexts where the two modal auxiliaries can also be used. It especially competes with could in the past time sphere, as shown in (11) and (12). Leclercq and Depraetere (2022) argue that the choice of could or be able to largely depends on whether the speaker intends to communicate an implicature of actualisation. While could remains entirely non-factual, be able to is associated with an implicature of actualisation: in (12), not only did the subject referents have the ‘opportunity’ to carry out a certain action, they actually performed this action. This implicature makes be able to pragmatically distinct from the auxiliary could.

Another pragmatic dimension in terms of which two (or more) constructions may differ is with respect to illocutionary acts. By means of illustration, we will discuss the use of insubordinate if-clauses (13) and compare them with conditional if-clauses (14) and imperative clauses (15).

A number of studies have established the exact formal and functional properties of insubordinate if-clauses (e.g., Kaltenböck 2016; Mato-Míguez 2016; Stirling 1998). The most striking feature, compared to the more regular conditional if-clauses, is that insubordinate if-clauses (IICs) “can be used on their own, with the apodosis left unexpressed” (Huddleston and Pullum 2002: 945; though examples such as Miss Spellman, if you could give me their names, then I will have them in for review (COCA) can be found). As a result, IICs express a rather low (if any) degree of conditionality and instead foreground a directive speech act.^[3] This construction thus has a pragmatic profile that is clearly distinct from conditional if-clauses. The question though is whether and how this directive speech act differs from that conventionally associated with imperative clauses (15). Interestingly, it appears that while imperatives are often used to express ‘orders’, ICCs most often express ‘requests’ (Mato-Míguez 2016: 303). These constructions are therefore pragmatically non-equivalent.

2.3 Differences in social meaning

Finally, the principle of no equivalence predicts that two constructions can be functionally distinguished in terms of social meaning. Social meaning is a notion extensively studied in variationist sociolinguistics, especially the third wave (Eckert 2012), where it is defined as the information drawn about speakers and their communicative practices (Hall-Lew et al. 2021). However, its incorporation into a Construction Grammar model has only been infrequently undertaken (Hollmann 2013; Hoffmann 2015; Morin 2023; Morin et al. 2020; Östman and Trousdale 2013), due to the general focus of Cognitive Linguistics on individual cognition rather than community convention and variation (Dąbrowska 2016). This relative neglect of social variability has led linguists to call for a widespread theoretical and empirical “social turn” in the discipline (Geeraerts 2016; Morin et al. 2024; Schmid 2016). More recently, Hoffmann (2022: 238) shows that a good starting point for a Construction Grammar explanation of sociolinguistic variation is to consider constructions as conveying social meaning in addition to semantic meaning. Take, for instance, the presence of/r/in word-final and pre-consonant positions (i.e., rhoticity) in varieties of New York English from the 1960s onwards, classically studied by Labov (1972), and represented as constructions by Hoffmann (2022: 238) in (16) and (17) below.

Based on these examples, Hoffmann argues that the variable pronunciation schema of (17), instantiated in words such as skirt (16), leads to the entrenchment and conventionalisation of two separate constructions, distinguished in terms of social information relating to ‘class’: one is associated with a rhotic pronunciation and the social meaning feature ‘upper class’ as in (16), while the other is associated with a non-rhotic pronunciation and the social meaning feature ‘working class’. Hoffmann thus operationalises the classic notions of the sociolinguistic variable and interchangeable sociolinguistic variants from variationist theory into a Construction Grammar model of linguistic knowledge, i.e., where “the variants are identical in reference or truth value, but opposed in their social and/or stylistic significance” (Labov 1972: 272).

In order to more systematically operationalise the social meaning of constructions, Morin (2023) proposes that this category should subsume the meso- and macro-levels of the context of language use delineated by Culpeper (2021). These levels encompass information relating to the non-exhaustive range of “activity types, frames, genres, discourses, social practices” for the meso-level and “ideologies, cultures, nationalities, genders, ages” for the macro-level (Culpeper 2021: 26). This framework is more comprehensive than the single example of a ‘stylistic aspect’ found in Goldberg (1995), namely ‘register’ (which is discussed in pragmatic rather than social terms). Thanks to this fine-grained classification, we are able to explain a much wider array of constructional alternations that are defined at least in part by their social conditioning. Take the simple examples of (18) and (19), which are discussed by Uhrig (2015) in his critical assessment of no synonymy.

In (18), the comparison of the spelling variants color/colour in very similar linguistic contexts (academic texts spelling out the shades of red and green that are needed to compose a new shade, yellow) strongly suggests that the alternatives, while sharing identical semantic meanings and lacking pragmatic meanings, are constrained by well-known social norms stemming from standard language institutions. Namely, the former orthography is sanctioned as the correct one in American English,^[4] while the latter is the correct one in British English.^[5] This is further evidenced by the distributions of the variants in COCA and the BNC, where color (124814 tokens) is over 25 times more frequent than colour (4861 tokens) in the American corpus, and colour (11135 tokens) is almost 100 times more frequent than color (112 tokens) in the British corpus. No equivalence predicts both the co-existence of these constructions across varieties as dialect-specific constructions, and the pre-emption of one in favour of the other within a single variety. This specific example showcases a macro-level feature (regional variation) as the main distinctive force between two individual constructions. Similarly, it is a social meaning feature, this time at the meso-level, which appears to govern the alternation between that and zero-subordinator constructions in (19). In their recent quantitative and qualitative study of this grammatical alternation, Gadanidis et al. (2021: 6) show that in addition to distributional differences,^[6] a crucial factor governing the choice of the variant is in terms of register, namely the formal and written registers in the case of that and the informal and spoken registers in the case of zero.

Foregrounding the previously underappreciated role of social information in the meaning of constructions is all the more necessary as it has in fact been cited as counter-evidence to the principle of no synonymy. Indeed, Uhrig (2015: 332) claims that sociolinguistic variants are “automatically excluded by [no synonymy]”, and argues that the principle is one of “no variation” (p. 331). Similarly, Gardner et al. (2021: 2) claim that language variation is not predicted by the constructionist view of isomorphism – that “if two grammatical forms exist, they must or should have a different meaning”. As shown in Leclercq and Morin (2023), however, this criticism relies on a conflation between synonymy avoidance and semantic synonymy avoidance, which may have been caused by informal definitions of the sociolinguistic variable, namely, “two or more ways of saying the same thing” (Sankoff 1980: 55). In the principle of no equivalence, social meaning is clearly highlighted as a component of linguistic function in its own right, and an important source of constructional distinction in alternation paradigms.

From the overview above, it may seem like two constructions typically differ only along one axis, be it semantic, pragmatic or social. We do not mean to claim that, however. Rather, constructions generally differ on various axes simultaneously. For instance, it has been thoroughly demonstrated that the dative alternation between the DITRANSITIVE construction in (20a) and the to-DATIVE construction in (20b) is largely determined by information structure, a pragmatic feature, with e.g., short, given and definite recipients eliciting the use of the DITRANSITIVE construction and long, new and indefinite recipients eliciting the to-DATIVE (Bresnan et al. 2007; Röthlisberger 2018a; Theijssen 2012). There are also social differences between these constructions, however, among the various varieties and registers of English (Engel et al. 2021; Röthlisberger et al. 2017). In addition, both constructions occasionally exhibit semantic differences (Röthlisberger 2018b: 47–51; Thompson and Koide 1987: 400). For one, the Ditransitive construction has been argued to imply closeness between agent and recipient, while the to-Dative implies distance. This would explain why (21a) is judged to be less acceptable than (21b) (examples taken from Kirsner 1986: 150, cited in Thompson and Koide 1987: 401).

The multifactorial nature of the dative alternation thus illustrates that a difference between two constructions on one axis in no way invalidates a difference on another.

4.1 Model design

With this agent-based simulation, we intend to show that effects of no equivalence can be captured by focusing on two basic maxims of language, viz. ‘optimal expressivity’ and ‘usage affects grammar’, with no further theoretical assumptions needed to explain it. The first maxim is optimal expressivity (Christiansen and Chater 2008; Goldberg 1995: 67–68; Gibson et al. 2019; Haspelmath 2023; Hawkins et al. 2018; Regier et al. 2015; Brochhagen and Boleda 2022; Zipf 1949), which refers to the efficiency trade-off between communicative pressures for linguistic informativeness and cognitive pressures for linguistic simplicity. That is, although a language user will seek to use a diverse enough range of words to express specific, informative meanings, they will also prefer to use an existing word to express a meaning, as long as this word is reasonably close to the meaning that they intend to express.^[11] Language users will only invent a new word when they have no words at their disposal that come sufficiently close to the meaning that they intend to express.

For example, imagine that you want to ask your interlocutor to hand you a specific drinking vessel. You may not have seen this particular drinking vessel before, but you do know a few words that you have heard to refer to similar objects in the past, such as cup, mug, beaker, pint etc. You compare the meaning that you intend to the meanings of these words, taking into account the semantic, pragmatic and social axes of these meanings, and find that the meaning that you associate with cup comes closest. You decide that this meaning of cup is also reasonably close to what you intend to express: it is optimally expressive. So you ask for the cup. By contrast, now imagine that you are part of the ghetto-style scene in Los Angeles in the second half of the 20th century, where a new cultural custom is emerging. This is the act of adding all sorts of ornaments to a car, thereby making it look overly flamboyant and perhaps a bit ridiculous. You could in principle use an existing English word for this act, such as to fancify, to embellish, to embroider, but none of those really fit what you intend to express: none of them is optimally expressive. In this particular case, the social dimensions of their meanings are especially ill-fitting. So you come up with a new word: to pimp (cf. De Pascale et al. 2022; Pijpops et al. 2023; Van de Velde and Zenner 2010). These examples of course focus on the intuitive case of words, but they should be taken as illustrative of constructions at large.

The second maxim is that usage affects grammar. This is the main assumption of usage-based linguistics, which holds that “grammar is the cognitive organization of one’s experience with language” (Bybee 2006: 711). That is, grammar is viewed as a dynamic cognitive system of emergent categories constantly updated by language use (Diessel 2019). In particular, agents will continuously update their meaning representations of constructions based on how they hear them being used. These are the only assumptions that we will build into the simulation. In all other design choices, we will strive to build the simplest model, in accordance with best practices in agent-based simulation (Landsbergen 2009: 18–19; van Trijp and Steels 2012: 9). If we can then show that no equivalence emerges in such a simple simulation, then we have clear evidence that these assumptions are necessary and sufficient to explain its existence in the real world.

The simulation, built in Python (van Rossum and Drake 2009), consists of a population of p agents. When a simulational run is executed, these agents will engage in a set number of interactions. For each interaction, two agents are chosen at random: one producer and one comprehender agent. A visualization of such an interaction can be found in Figure 1. The producer is assigned a meaning to be expressed. Meanings are operationalized as n-dimensional vectors, as is common in usage-based linguistics (e.g., Geeraerts et al. 2023; Levshina and Heylen 2014; Pijpops et al. 2021). These vectors are meant as representations of the full meaning of constructions, including not just semantic meaning, but pragmatic and social meaning as well. Since it is only possible to visualise a finite vector space of maximally three dimensions, the coordinates of the vectors range between 0 and 10 and n was set to 3. The producer agent is assigned a random position in this meaning space to express to the comprehender. In the example in Figure 1, this is the meaning [1.8 1 0.8].

Figure 1:

Potential behaviours of the agents during the simulation (from Leclercq et al. 2025a).

The producer will then turn to their constructicon, which contains constructions, i.e., pairings of constructional forms, represented as strings, and meanings, represented as vectors. They will calculate the distances between the meaning to be expressed and the meanings of these constructions, using Euclidean distance as in Equation (1). The producer then picks the closest construction, e.g., the construction with form “A” in Figure 1.^[12] This is where optimal expressivity comes in. If the meaning of this construction is reasonably close to the meaning that the agent intends to express, that is, if it is closer than threshold t, then the producer will decide to use the construction. This is the case in Scenario 1 in Figure 1, which corresponds to the cup-example introduced above. If the producer knows no construction whose meaning comes closer than the threshold t, then the producer will invent a new construction with a unique form specifically to express this new meaning. This corresponds to Scenario 2 in Figure 1 and to the pimp-example.^[13] If the producer’s constructicon is still completely empty, the producer will also invent a new form. This only ever occurs at the very start of the simulation, when an agent is selected as speaker that has yet to act as hearer. These newly invented forms are always unique: it is assured that the same constructional form is not invented multiple times during the same simulational run. The producer then expresses the constructional form to the comprehender and points at the intended meaning.

Equation (1): Calculation of the Euclidean distance between two vectors P ₁and Q ₂ with respective coordinates x ₁,y ₁, z ₁ and x ₂, y ₂, z ₂.

This is when the second maxim, viz. usage affects grammar, comes in. Specifically, upon hearing the producer, the comprehender will adapt their constructicon in some way, be it slightly or strongly. If the comprehender has heard the constructional form before, e.g., “A” in Figure 1, they will update its associated meaning. Specifically, they will calculate its new meaning as the average of its existing meaning, e.g., [2.2 1.2 0.4] in Figure 1, and the new intended meaning, e.g., [1.8 1 0.8], which would result in [2 1.1 0.6]. If the comprehender has never heard the constructional form before, they add it to their constructicon, with the intended meaning as its constructional meaning. This is always the case in Scenario 2, since it is ensured that the same forms are never invented multiple times. At this point, the interaction ends and a new producer and comprehender are randomly selected.

As the simulation runs and the number of interactions increases, we keep track of the constructicons of all agents by recording two main measures at 1000 measure points: for instance, for 10,000 interactions, we record measures every 10 interactions, and for 100,000 interactions, every 100 interactions. Specifically, for reasons that will be explained shortly, we calculate the following processes over time:

1. The degree of convergence (K) of meanings of constructions shared by agents, in order to assess conventionalisation. To measure this, we compute the pairwise Euclidean distances between the prototypical meanings of each construction across all agents who have it stored in their constructicon. We then average the pairwise distances for each construction, and then average the averages across constructions, as in Equation (2). Finally, we subtract this result from the number 10 so that a higher value will indicate a higher degree of convergence, and a lower value will indicate a lower degree of convergence.

   K = 10 −   1 C ∑   c ∈ C 1 P c   ·   P c − 1 / 2 ∑ p 1   <   p 2 ∈ P c d p 1 , p 2

   Equation (2): Calculation of convergence K, with C: all unique constructions, c: one unique construction, Pc: meaning positions of c in the constructicons where c appears, d(p1,p2): distance between two positions, <: precedence relationship.

2. The equitability (E) of the meaning space distribution across constructions in the constructicons. To measure this, we compute the pairwise distances between the meanings of all constructions in each agent’s constructicon. Then, we select the smallest distance for each construction, and we calculate the range of these distances by subtracting the minimum from the maximum value. Finally, we sum the ranges and we divide the sum by the total number of agents, as in Equation (3). Finally, as for K, we subtract this result from the number 10 so that a higher value will indicate a higher equitability, and a lower value will indicate a less equitable partition of the meaning space across constructions in the community.

   E = 10 −   1 N ∑ i = 1 N ma x m 1   <   m 2 ∈ M i d m 1 , m 2 − mi n m 1   <   m 2 ∈ Mi d m 1 , m 2

   Equation (3): Equitability measure E, with N: total number of agents, Mi: meanings of constructions in a constructicon, m: meaning of one construction in the constructicon, d(m1,m2): distance between two positions, <: precedence relationship.

Both measures are of particular interest for a study of no equivalence. The first measure is crucial as it enables us to make sure that the constructions used by agents conventionalise over time (with a higher degree of convergence indexing a higher degree of conventionality). Given that conventionality is a key factor in no equivalence, a high degree of convergence is required before we can continue testing the principle. The second measure is also crucial for a study of no equivalence, as it tells us to what degree there is overlap in meaning areas covered by constructions. Two types of results are possible, as illustrated in 2D space in the toy example of Figure 2. If equitability is lower, constructional meanings cluster together in an unequal partition of the meaning space (left plot). If equitability is higher, constructional meanings are more evenly spread across the partition space (right plot).

Figure 2:

Representing two possible results of equitability measuring – 30 data points (from Leclercq et al. 2025b).

If we find an increase of equitability of the constructicons of agents, this is a clear indication that constructions partition the totality of the functional space and carve individual functional niches that are not impinged on by other constructions, and the principle is validated. If we do not find such an increase, the principle is falsified. In order to assess this more intuitively, we will also visualise constructicons from the end of the simulations as 3-dimensional spaces to see whether constructions cluster.

The design of the simulation bears some similarity to the Naming Game (Steels 1995, cf. Wellens 2012: 35–36 for a concise overview), but is crucially different in that the meaning space is continuous and its division is not predetermined. This is important, as it allows for the division of the meaning space to shift as more forms are introduced. The Principle of No Equivalence would then entail that the meanings of constructions continuously tend to move away from each other, yielding an increase in equitability.

4.2 Results and discussion

The parameter settings of the main model of this study are the following:

Population size n: 10

Interaction number i: 10,000

Meaning distance threshold t: 3

These settings were chosen as a means to collect a sufficiently diverse but manageable number of data points for visualisation purposes. Variations of this model (with 100 agents, 100,000 interactions, and/or a 5-unit meaning distance threshold) are appended to the paper to confirm that the effects are consistent across parameter settings.

We ran the model in a batch of 100 simulations, keeping track of the minimum, average, and maximum values of the sizes of the constructicons at the end of each simulation, the convergence measure K, and the equitability measure E. Regarding the size of the constructicons, we find that agents in the community create and conventionalise an average of 99 constructions, with a minimum size of 85 and a maximum size of 114.

In Figure 3, we visualise K at 1000 measure points. Overall, the simulations yield very similar results with little variation between the minimum, average, and maximum values. In general, it appears that constructional meaning convergence troughs at the very start of the simulation, roughly over the first few hundred interactions, and then sharply rises to stabilise at more than 9.95 by the 2000th interaction. We thus find a strong meaning convergence effect across the constructicons in the community.

Figure 3:

Constructional meaning convergence in community over time.

The evolution of K over time indicates that agents very quickly align their constructions with one another as they acquire and use them in the community, an expected result of usage-based learning and co-adaptation. This result is valuable in that it reflects patterns of conventionalisation in language, and that the arena of use of the phenomena we study is the community and not just the individual. We therefore have a good starting point for a study of no equivalence because the scope of this principle is the construction at the level of individual entrenchment and population-wide conventions, rather than just individual usage (see Section 3).

Having established that agents quickly conventionalise constructions and their meanings across their individual grammars, we now turn to examining how exactly these constructions are distributed across the meaning space. We do this in two steps.

First, we visualise E at 1000 measure points in Figure 4. By contrast with Figure 3, the simulations display a larger amount of variation between the minimum, maximum, and average values. However, we find a similar general pattern with a trough of values in the initial stages of the simulation and a sharp and continued increase followed by stabilisation in the later stages.^[14] Specifically, agents initially display only a moderate degree of equitability in the partitioning of the meaning space in their constructicons, and this degree has increased sharply, roughly by two units, by the final interactions.

Figure 4:

Meaning space equitability in community grammar over time.

The figure shows that agents tend to shape their constructicons in a way that aims to equitably partition the meaning space with a finite set of constructions. On the one hand, this result may be interpreted as falling out of the maxim of ‘optimal expressivity’, represented in the model by the meaning distance threshold for producer agents. On the other hand, it was not predicted by the maxim of ‘usage affects grammar’, which might well have yielded an opposite result. Given the instruction that comprehender agents add every (new) construction to their grammatical knowledge regardless of how similar their meaning is to previously stored constructions, they could easily have stored an open-ended amount of redundant constructions skewing the partition of the meaning space. Against these odds, the figure indicates a notable, robust increase of the equitability of their constructicons. This is clear and strong evidence for the principle of no equivalence. Indeed, equitability in the meaning distance between constructions precludes the existence of two fully overlapping constructions with identical functions. In other words, if the constructions are spread equitably, there can be no equivalent constructions. We further explore this finding in the second step below.

To get a better sense of the grammatical systems of agents at the end-point of the simulations, in Figure 5 we visualise the final constructicon of a random agent from a random simulation.^[15] In particular, we plot the meanings of all constructions in the 3-dimensional vector space. The 3 dimensions index the three vector levels of the meaning space operationalised in our simulation: you could imagine them to represent semantic, pragmatic, and social meaning. In order to better visualise three dimensions on a two-dimensional sheet of paper, we furthermore use an RGB colour scheme to show how far along the three dimensions each construction is: red corresponds to dimension 1, green to dimension 2 and blue to dimension 3. Constructions that are close in space are also closer in colour.

Figure 5:

Random constructicon of agent 1 (x = Dim1, red colour gradient; y = dim 2, green colour gradient; z = dim 3, blue colour gradient).

As expected based on Figure 4, with Equitability at an all-time high, we indeed find that there are no real voids in the meaning space, and also no places where several constructions have the same colour. All competing constructions differ in form, and they are all distinguished along one or more of the meaning axes. Some constructions will be functionally very distinct (consider the red and blue points at the opposite corners of the meaning space), while other constructions will be substantially more similar (e.g., the points displaying shades of red, pink, and purple at the center of the plot). Crucially, none of them ever overlap in all three dimensions at the same time. In sum, the figure translates no equivalence in visual terms.